ANGLES AND ARCS 199 1. Draw an arc of a circle. Draw radii from its end points to the center. The arc is said to subtend the angle between the radii. 2. Draw a circle and two diameters at right angles. Into how many parts is the circle divided? What angle does each subtend? The size of an arc is often expressed in degrees, an arc of one degree being an arc which subtends an angle of 1°. 3. By use of a protractor and a circle of radius 3 in., draw arcs of 15°; 25°; 90°; 110°; 160°; 180°. 4. Draw arcs of 190°; 210°; 250°; 270°; 300°; 360°. What part of an entire circle is each arc you have drawn? 5. What is the length of a circle of radius 5 feet? What is the length in feet of a 180-degree arc of this circle? Of a 30-degree arc? Of a 270-degree arc? 6. The railway curve shown in the picture is a 60-degree arc of a circle. The rails are standard gage (4 ft. 8 in. apart); find the length of each rail in this arc. 500' 7. A fountain is surrounded by 3 equal flower-beds as shown in the picture; how many degrees are there in the whole angle at the center of the fountain? 8. What part of 360° is the angle belonging to one flower-bed? How many degrees are there in each of these angles? 9. How many degrees are there in the arc belonging to each flower-bed? 200 NUMBERS REPRESENTED BY LETTERS Oral. 1. John had a certain number of marbles. Represent the number by n. What represents the number, if he buys 2 more? 2. What represents the number if he gives away 4? 3. James has twice as many as John had at first; how many has James? 4. Frank has 2 more than twice this number; how many has Frank? 5. Tom has one less than 3 times n marbles; how many has Tom? 6. When John had n and James 2 n, how many did both have? 7. John gave of his to Tom. How many had Tom then? How many had John left? 8. If the number which John had was 12, how many had Frank? James? Tom? Written. State the following in figures or letters, using signs for operations: 9. One hundred is forty more than sixty. 10. The length (1) of a box is twice its breadth (b). II. The number of quarts (q) in a certain number (g) of gallons. 12. A man is y years old. His father is twice as old. 13. The sum of the ages of father and son in Exercise 12 is 60 years. 14. A statue is 3 times as high as its pedestal, the height of the pedestal being h. 15. The number denoted by x is to be subtracted from that denoted by y. 16. A pair of gloves costs c cents; what would the cost be, if the price were raised 5 cents? NUMBERS REPRESENTED BY LETTERS 201 1. If there are 8 books on one shelf and 12 books on another, how many books are there on both shelves? If there are a books on one shelf and b books on another, how many books are there on both shelves? Write this sum. 2. What is the value of a+b when a 6 and b When a 8 and b=20? = When a 16 and b = 14? = 14? 3. A merchant sold goods which cost $6 for $8; how many dollars did he gain? If goods costing $b were sold at a profit for $a, how many dollars were gained? Write an expression indicating this profit. = 4. What is the value of ab when a 25 and b = 10? When a 15 and b = 7? When a = 30 and b = 14? 5. If a barrel of apples costs $3, what will 3 barrels cost? 4 barrels? 12 barrels? n barrels? 6. If there are 4 rows of apple-trees in an orchard and 8 trees in each row, how many trees are there in the orchard? How many trees are there in an orchard with b rows of 8 trees each? How many trees are there in an orchard with b rows of a trees each? How is the product of a and b written? 7. How many tons of coal at $5 a ton can be bought for $25? How many tons at b dollars a ton can be bought for a dollars? The operation a divided by b is written ab org. 8. What is the value of a when a = 30 and b = 6? When a 42 and b 7? When a = 56 and b = 7? Find the value of each expression when a = 2, b = 1, c = 3, 202 PROBLEMS SOLVED BY USE OF LETTERS 1. A contractor, wishing to dig a cistern, finds it most convenient to make it 5 ft. square; how deep must it be dug to contain 750 gal.? (Use 1 cu. ft. = 7 gal.) PLAN. 1. 750 ÷ 7 = −, the number of cu. ft. required. 2. Let d stand for the depth. 3. 5 X 5 X d = the number of cu. ft. 4. Therefore d = —, the number of ft. of depth required. 2. The metal types used in printing are made of 4 parts by weight of lead to 1 part of antimony; how much lead is there in 86 lb. of type? Antimony? PLAN. 1. Let a stand for the number of pounds of antimony in the type. 2. The number of pounds of lead is 4 times -. 3. A rectangular park is twice as long as it is wide and the distance around it (perimeter) is 18 miles; find its length and breadth in miles. PLAN. 1. Let x stand for the number of miles in the width of the park. 2. The number of miles in the length is X. 3. The perimeter is 2x + 4 x = = and the breadth = miles. 4. The perimeter of a rectangular farm is 1 The length is of a mile more than the breadth; find its dimensions. 5. A man left of his estate to his children and divided the remainder equally among 4 charitable institutions, each receiving $6,000; what was the value of his estate? 6. A newsboy delivers 75 papers per day. He delivers twice as many in the morning as in the evening; how many does he deliver in the morning? 7. A man owned of a store; he sold of his share for $3,000; find the value of the store. REVIEW AND SUMMARY 203 1. How many degrees are there in the sum of the angles of a triangle? 2. What is the sum of the acute angles of a right-angled triangle? 3. What is an acute angle? An obtuse angle? How many obtuse angles can a triangle have? 4. What is a cash account? What are the names of the two sides of the account? 5. On which side of the account are items of money expended recorded? Of money received? How is cash thought of, so that this may be done? 6. What is meant by balancing an account? 7. Make out a cash account for the following: Mr. Roe received $35 interest Jan. 1, 1904, for some money lent; $105, January 15, for the sale of some produce; and $18, January 24, from the rent of a house. He paid out $15 for clothing, Jan. 5, $10 for provisions, Jan. 13, and $50 for a wagon, Jan. 30. 8. If n represents a number, what does 5 n represent? 9. Write n, 2n, 3 n, to 10 n, and write the values of these numbers when n = 2, 3, 4, 5, . . . . 10. Of what are the numbers of the first set multiples? Of the second set? Of the third? Of the eighth? 10. If n represents Henry's age, what represents his age 5 years ago? What represents his age 5 years from now? 11. If Henry is x years of age and George is y years of age, what represents the sum of their ages?of the sum of their ages? 12. If the adjacent sides of a rectangle are x and y, what represents the perimeter of the rectangle? 13. Belle had a apples and Susie half as many. How many had Susie? |